A130780 -id:A130780 - cp's OEIS frontend

A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      13         15         18         20           22
   ------------------------------------------------------------------
    (2,1)  (6,5,2)    (10,5)     (12,6)     (12,7,1)     (12,8,2)
           (6,4,2,1)  (6,4,3,2)  (8,7,3)    (8,5,4,3)    (8,6,5,3)
                      (6,5,3,1)  (8,5,3,2)  (8,6,4,2)    (8,7,5,2)
                                 (8,6,3,1)  (8,7,4,1)    (12,7,2,1)
                                            (8,6,3,2,1)  (8,6,4,3,1)
                                                         (8,7,4,2,1)

This is the strict case of A045931, ranked by A350848 (zeros of A350941).
The conjugate version is A239241, non-strict A045931 (ranked by A325698).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, ranked by A349157, strict A352131.
- A277103, ranked by A350944.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A171967 Number of partitions of n with distinct numbers of odd and even parts.

Original entry on oeis.org

0, 1, 2, 2, 5, 5, 10, 12, 20, 25, 37, 49, 68, 90, 119, 158, 206, 269, 344, 446, 565, 722, 908, 1148, 1435, 1795, 2229, 2765, 3416, 4204, 5164, 6315, 7717, 9380, 11406, 13793, 16692, 20093, 24203, 29012, 34799, 41552, 49636, 59059, 70279, 83341, 98822

Offset: 0

Reinhard Zumkeller, Jan 21 2010

nonn

a(n) = A000041(n) - A045931(n) = A108949(n) + A108950(n).

a(n) = Sum_{k<>0} A240009(n,k). - Alois P. Heinz, Mar 30 2014

Alois P. Heinz, Table of n, a(n) for n = 0..3500

Cf. A130780, A171966.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t<>0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014

$RecursionLimit = 1000; b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t != 0, 1, 0], If[i < 1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 2    7        9        13        14         15         16
   --------------------------------------------------------------------
    (2)  (6,1)    (8,1)    (12,1)    (14)       (14,1)     (16)
         (4,2,1)  (4,3,2)  (6,4,3)   (6,5,3)    (6,5,4)    (8,5,3)
                  (6,2,1)  (8,3,2)   (10,3,1)   (8,4,3)    (12,3,1)
                           (10,2,1)  (6,4,3,1)  (10,3,2)   (6,5,4,1)
                                     (8,3,2,1)  (12,2,1)   (8,4,3,1)
                                                (6,5,3,1)  (10,3,2,1)
                                                           (6,4,3,2,1)

This is the strict case of A277579, ranked by A350943 (zeros of A350942).
The conjugate version is A352131, non-strict A277579 (ranked by A349157).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944, strict new.
- A350948, ranked by A350945, strict new.
There are three double-pairings of statistics:
- A351976, ranked by A350949, strict A010054?
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980. strict A014105?
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      10         14         18         21             24
   ----------------------------------------------------------------------
    (2,1)  (6,4)      (8,6)      (10,8)     (11,10)        (8,7,5,4)
           (4,3,2,1)  (5,4,3,2)  (6,5,4,3)  (8,6,4,3)      (9,8,4,3)
                      (6,5,2,1)  (7,6,3,2)  (8,7,4,2)      (10,8,4,2)
                                 (8,7,2,1)  (10,8,2,1)     (10,9,3,2)
                                            (6,5,4,3,2,1)  (11,10,2,1)
                                                           (8,6,4,3,2,1)

This is the strict case of A277579, ranked by A349157 (zeros of A350849).
The conjugate version is A352130, non-strict A277579 (ranked by A350943).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350942, A350950, A350951.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,30}]

A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 15, 21, 27, 38, 47, 63, 79, 106, 130, 170, 209, 272, 330, 422, 512, 653, 784, 986, 1183, 1482, 1765, 2191, 2604, 3218, 3804, 4666, 5504, 6726, 7898, 9592, 11240, 13602, 15880, 19122, 22277, 26733, 31048, 37102, 43003, 51232, 59220

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(7) = 8 partitions: (7), (511), (421), (331), (322), (31111), (22111), (1111111). Missing are: (61), (52), (43), (4111), (3211), (2221), (211111).

Even- and odd-indexed terms are A006330 and A001523 respectively, which add up to A000712.

Cf. A000898, A001405, A026010, A045931, A058696, A063886, A097613, A130780, A171966, A239241, A300788, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&]],{n,0,50}]

A325699 Number of distinct even prime indices of n minus the number of distinct odd prime indices of n.

Original entry on oeis.org

0, -1, 1, -1, -1, 0, 1, -1, 1, -2, -1, 0, 1, 0, 0, -1, -1, 0, 1, -2, 2, -2, -1, 0, -1, 0, 1, 0, 1, -1, -1, -1, 0, -2, 0, 0, 1, 0, 2, -2, -1, 1, 1, -2, 0, -2, -1, 0, 1, -2, 0, 0, 1, 0, -2, 0, 2, 0, -1, -1, 1, -2, 2, -1, 0, -1, -1, -2, 0, -1, 1, 0, -1, 0, 0, 0

Offset: 1

Gus Wiseman, May 17 2019

sign

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

a(n) = A324967(n) - A324966(n).

Cf. A000712, A001221, A026010, A045931, A063886, A112798, A130780, A171966, A241638, A325698, A325700.

Table[Total[(-1)^PrimePi/@First/@If[n==1,{},FactorInteger[n]]],{n,100}]

G.f.: Sum_{k>=1} (-1)^k * x^prime(k) / (1 - x^prime(k)). - Ilya Gutkovskiy, Feb 12 2020

Additive with a(p^e) = (-1)^primepi(p). - Amiram Eldar, Jun 17 2024

A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 2, 2, 5, 2, 5, 4, 6, 7, 7, 8, 8, 9, 9, 13, 9, 14, 12, 20, 13, 25, 17, 33, 23, 40, 26, 50, 33, 59, 39, 68, 45, 84, 58, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      18         22          28           31              32
   -----------------------------------------------------------------------
    (2,1)  (8,5,3,2)  (8,6,5,3)   (12,7,5,4)   (10,7,5,4,3,2)  (12,8,7,5)
           (8,6,3,1)  (8,7,5,2)   (12,8,5,3)   (10,7,6,5,2,1)  (12,9,7,4)
                      (12,7,2,1)  (12,9,5,2)   (10,8,5,4,3,1)  (16,9,4,3)
                                  (16,9,2,1)   (10,9,6,3,2,1)  (12,10,7,3)
                                  (12,10,5,1)                  (12,11,7,2)
                                                               (16,11,4,1)

The first condition is A239241, non-strict A045931 (ranked by A325698).
This is the strict version of A351977, ranked by A350946.
The second condition is A352129, non-strict A045931 (ranked by A350848).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, strict A352131.
- A277103, ranked by A350944, strict A000700.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are two other double-pairings of statistics:
- A351976, ranked by A350949.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A014105, A088218, A098123, A195017, A236559, A236914, A241638, A325700, A350839, A350941.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[#,?EvenQ]&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

cp's OEIS Frontend

A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

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A171967 Number of partitions of n with distinct numbers of odd and even parts.

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A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

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A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

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A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

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A325699 Number of distinct even prime indices of n minus the number of distinct odd prime indices of n.

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Formula

A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

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Mathematica

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

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